Density: averaging always finds another rational in between

0:00One and two. Is there a number between them? Three halves, and you did not have to search for it. It is the obvious occupant of an obvious gap, so the question is not interesting yet. Make it harder. What sits between one and three halves? The gap is half as wide now. Between one and five quarters? Half again. Keep going and the gap gets smaller than anything you can point at, and the eye stops being useful.

0:33So take that second gap, one and three halves, and instead of spotting the answer, produce it. Add the two together and halve the result. One plus three halves is five halves. Half of five halves is five quarters. Five quarters. And look at where it sits: between one and three halves, and equal to neither of them. Now notice what you did not do. You did not examine the interval. You did not ask how much room was inside it. You performed the same two operations you would have performed on any other pair, and the answer came out.

1:13That is the average, and for this topic it is worth thinking of it as a machine. Two inputs, one output. Feed in a and b. Out comes a plus b, over two. Two things have to be true for that machine to be worth anything, and neither is obvious. The output has to be a fraction, because if averaging two fractions could produce something that is not one, the machine would be answering a different question.

1:43And the output has to land strictly inside. If it could land on an endpoint, it would sometimes hand you back a number you already had. Both are checkable, and neither is free. Take the first one. Write the two inputs as whole numbers over whole numbers: p over q, and r over s. Do the addition by hand and the top becomes p times s, plus r times q, over q times s. Then halve it, and the bottom becomes two times q times s.

2:21Look at what came out. The top is a whole number, because whole numbers added and multiplied stay whole. The bottom is a whole number too, and it is not nought, because q is not nought, s is not nought, and two is not nought. Twenty nine thousand two hundred and forty one averages were built this way, out of integer parts. Not one of them came out with a nought underneath.

2:50Now the second thing. Why can the output not land on an end? Here is the cleanest way to see it. Average a with itself and you get a. Average b with itself and you get b. So the machine returns an endpoint exactly when you feed it the same number twice. Feed it two different numbers and it cannot. One hundred and fifty three pairs of distinct numbers, every one of them averaged: not one output landed outside its pair, and not one landed on an end.

3:25And that is not automatic. Divide the sum by three instead of two, and you fall outside the pair on twenty three of those same one hundred and fifty three. Here is what makes the machine worth building. Its output is the same kind of thing as its input, so you can feed it straight back in. Start with one and two. Out comes three halves. Feed in one and three halves: five quarters. Again: nine eighths. Again: seventeen sixteenths.

4:00Look at the pattern. The bottoms are doubling, and the tops are always exactly one more than the bottom. Run it thirty times and the gap left over is one over two to the thirtieth. Not one of those thirty rounds landed on one, and not one went past it. The gap halves forever and never closes. Which is why infinitely many is not a separate marvel. It is just what the machine does when you leave it running.

4:32There is a second route, and it is the one worth having. Instead of averaging, rewrite both ends over a single cutting of the unit, and then look at the whole numbers left on top. Take minus a half and a quarter. Over quarters they read minus two and one, and the whole numbers strictly between minus two and one are minus one and nought. Two of them. If you wanted three, quarters are not enough. Go to eighths. The ends read minus four and two, and now you have minus three, minus two, minus one, nought, and one. Five candidates for the three you wanted.

5:16So how fine does the cutting have to be? Count it, rather than guess. If the two tops are whole numbers, the whole numbers strictly between them come to exactly one less than the gap. Five hundred and seventy four pairs were checked and that held on every one. So for five numbers you need a gap of at least six. Here it is in use. Seven twelfths and five sixths. Over twelfths the tops read seven and ten, a gap of three, which leaves two. Over twenty fourths they read fourteen and twenty, a gap of six, which leaves exactly five.

5:56And one warning that is easy to walk into. A cutting has to be able to write both ends, not merely have room between them. Fifths leave four whole numbers sitting inside minus a half and a quarter, and neither end can be written over fifths, so not one of those four is reachable. Both routes give you numbers in between. They do not give you the same numbers. Run the machine three times on one and two and you get three halves, five quarters, nine eighths. All three sit in the left half, bunched towards the end you kept feeding back in, and the widest they spread is three eighths.

6:40Take the single cutting instead. Quarters, between one and two, gives five quarters, six quarters, seven quarters. Spread over half the interval, and evenly, each one a quarter from the next. The machine proves that numbers are in there. The cutting arranges them. One more case, because it is the one that trips people up. Three point one four one five, and three point one four one six. Consecutive, surely. Nothing between them.

7:15They are consecutive ten thousandths. That is not the same as consecutive. Cut the unit into ten thousandths and there is genuinely no whole number left between the two tops. Add one more decimal place and there are nine of them, starting at three point one four one five one. And the machine needs to be told none of that. Feed it the two decimals and it hands back their average, strictly inside, without ever asking how finely to cut.

7:50So between any two of these numbers there is another one. That property has a name. Dense. And here is the trap the whole idea rests on. Dense does not mean no gaps. Watch. Take all these numbers and deliberately throw one away. A half, say. Is what is left still dense? Take any two of them and average them. If the answer is not the number you threw away, you are finished. If it is, average again on one side, and now it cannot be, because it is strictly on one side of it.

8:26One hundred and thirty six pairs, and every single one of them was served. The plain average handed back the missing number on two of those pairs, and the second step rescued both. Read that again, because it is the whole point. A set can pass the betweenness test on every pair it has, and still be missing a point. And the missing one is sitting strictly inside seventy two of those pairs, while the test says nothing at all about it.

9:00Dense is a statement about pairs. It is not a statement about points. So here is the question, and this is where we stop. These numbers crowd the line so thickly that between any two of them lie infinitely many more. It certainly feels as though there can be nothing left over for anything else to occupy. Is that feeling right? Do they fill the line?